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Theorem sbcrext 2811
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcrext ((A 𝑉 yA) → ([A / x]y B φy B [A / x]φ))
Distinct variable groups:   x,y   x,B
Allowed substitution hints:   φ(x,y)   A(x,y)   B(y)   𝑉(x,y)

Proof of Theorem sbcrext
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbcco 2760 . 2 ([A / z][z / x]y B φ[A / x]y B φ)
2 simpl 102 . . 3 ((A 𝑉 yA) → A 𝑉)
3 sbsbc 2743 . . . . 5 ([z / x]y B φ[z / x]y B φ)
4 nfcv 2160 . . . . . . 7 xB
5 nfs1v 1796 . . . . . . 7 x[z / x]φ
64, 5nfrexxy 2337 . . . . . 6 xy B [z / x]φ
7 sbequ12 1636 . . . . . . 7 (x = z → (φ ↔ [z / x]φ))
87rexbidv 2303 . . . . . 6 (x = z → (y B φy B [z / x]φ))
96, 8sbie 1656 . . . . 5 ([z / x]y B φy B [z / x]φ)
103, 9bitr3i 175 . . . 4 ([z / x]y B φy B [z / x]φ)
11 nfnfc1 2163 . . . . . . 7 yyA
12 nfcvd 2161 . . . . . . . 8 (yAyz)
13 id 19 . . . . . . . 8 (yAyA)
1412, 13nfeqd 2174 . . . . . . 7 (yA → Ⅎy z = A)
1511, 14nfan1 1440 . . . . . 6 y(yA z = A)
16 dfsbcq2 2742 . . . . . . 7 (z = A → ([z / x]φ[A / x]φ))
1716adantl 262 . . . . . 6 ((yA z = A) → ([z / x]φ[A / x]φ))
1815, 17rexbid 2301 . . . . 5 ((yA z = A) → (y B [z / x]φy B [A / x]φ))
1918adantll 448 . . . 4 (((A 𝑉 yA) z = A) → (y B [z / x]φy B [A / x]φ))
2010, 19syl5bb 181 . . 3 (((A 𝑉 yA) z = A) → ([z / x]y B φy B [A / x]φ))
212, 20sbcied 2774 . 2 ((A 𝑉 yA) → ([A / z][z / x]y B φy B [A / x]φ))
221, 21syl5bbr 183 1 ((A 𝑉 yA) → ([A / x]y B φy B [A / x]φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1373   wcel 1375  [wsb 1627  wnfc 2147  wrex 2283  [wsbc 2739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2288  df-v 2535  df-sbc 2740
This theorem is referenced by: (None)
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